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TEACHING AND LEARNING PORTFOLIO

by

Patrick Rault

Last Updated: August 6th, 2008

This portfolio submitted in partial fulfillment of the requirements for theDelta Certificate in Research, Teaching, and Learning.

Delta Program in Research, Teaching, and LearningUniversity of Wisconsin-Madison

The Delta Program in Research, Teaching, and Learning is a project of the Center of the Integration of Research, Teaching, and Learning (CIRTL—Grant No. 0227592). CIRTL is a National Science Foundation sponsored initiative committed to developing and supporting a learning community of STEM faculty, post-docs, graduate students, and staff who are dedicated to implementing and advancing effective teaching practices for diverse student audiences. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

For more information, please call us at 608-261-1180 or visit http://www.delta.wisc.edu.

Teaching Portfolio ii P. X. Rault

TEACHING AND LEARNING PORTFOLIO

Patrick Rault

Department of MathematicsUniversity of Wisconsin Madison

[email protected]

August, 2008

Teaching Portfolio iii P. X. Rault

Table of Contents

TABLE OF CONTENTS ............................................................................................................ iv

OVERVIEW ................................................................................................................................... iv About this document.

INTRODUCTION: THE DELTA PROGRAM .......................................................................... 1 My background in math and science education.

TEACHING AND LEARNING TIMELINE ............................................................................... 2 A guide of my courses and activities to aid the reader.

TEACHING AND LEARNING PHILOSOPHY ......................................................................... 3 My philosophy about teaching and learning, which summarizes my development

as a teacher.

RESEARCH STATEMENT: MATHEMATICS EDUCATION ............................................ 6 What I would like to do as a postdoc.

REFLECTIONS ON TEACHING-AS-RESEARCH ................................................................ 11 Applying educational techniques.

REFLECTIONS ON THE BEAUTY OF MATH ...................................................................... 15 Mathematical motivation.

REFLECTIONS ON UNDERGRADUATE MATH PROJECTS AND RESEARCH ........... 19 Working with undergraduate students.

REFLECTIONS ON APPROACHABILITY ............................................................................ 23 How I interact with students.

APPENDIX 1 ................................................................................................................................. 26 Long artifacts concerning Teaching-as-Research.

APPENDIX 2 ................................................................................................................................. 68 Long artifacts concerning the Beauty of Math.

OverviewThis portfolio is a working document which catalogs my development as a

teacher. This portfolio serves two purposes:1) To record and reflect upon my teaching experiences to help me develop as a

teacher.2) To demonstrate to the reader what kind of teacher I am.

Teaching Portfolio iv P. X. Rault

Introduction: The Delta Programin Research, Teaching, and Learning

As a doctoral student at the University of Wisconsin at Madison I pursued a certificate in Research, Teaching, and Learning in the Delta Program. This subset of the national Center of the Integration of Research, Teaching, and Learning (CIRTL) program is a “community for faculty, academic staff, post-docs, and graduate students that will help current and future faculty succeed in the changing landscape of science, engineering, and math higher education.“ For more information, see www.delta.wisc.edu

The three founding pillars of the Delta Program are:1) Teaching-as-research “involves the deliberate, systematic, and reflective use of research

methods to develop and implement teaching practices that advance the learning experiences and learning outcomes of students/participants and teachers/facilitators.” My first reflection, on page 11, describes my experience with Teaching-as-Research.

2) Learning-through-Diversity: “Faculty and students bring an array of experiences, backgrounds, and skills to the teaching and learning process. Effective teaching capitalizes on these rich resources to the benefit of all, which we call Learning-Through-Diversity.” My participation in the Wisconsin Emerging Scholars (WES) program, discussed in my reflection on Teaching-as-Research (page 11) was my first experience with Learning-through-Diversity. Later, I had students work on and present projects to learn from each others' diverse ideas (artifact 12, page 19 reflection).

3) Learning Communities “bring people together for shared learning, discovery, and the generation of knowledge.“ As discussed previously, the WES program is an active-learning community. Artifact 8 (page 69) and the accompanying reflection (page 15) concern a slideshow I created to get students in the mood for this environment. Additionally, my internship project concerned the development of a learning community in a large-lecture environment; see artifact 6 on page 45 and the accompanying reflection on page 11.

Receipt of a Delta Certificate signifies completion of the Delta program of study:1) Two graduate courses on teaching and learning.

Diversity in the College Classroom: "Participants will take a critical yet practical look at how we define diversity and for what purposes, and discuss the ways different definitions of diversity might influence what and how we teach our disciplinary topics."

The College Classroom: "After completing this course, participants will be active participants in the interdisciplinary learning community that develops within the course and outside of it, know how to create an inclusive classroom environment that engages all learners, and use TAR in future classrooms of their own."

2) Participation in the Delta Learning Community I worked with Delta staff member Chris Pfund and math professor Bob Wilson to design

a math department TA training based upon the Delta program. For a discussion of this project, see my reflection on Teaching-as-Research (page 11) and artifact 7 (page 54).

3) Completion of a Teaching and Learning Internship I created and implemented a study to compare certain Large Lecture Techniques which

increase student-teacher interaction. This study was presented at the Joint Meetings of the AMS and MAA in January 2008 and in the UW Teaching Improvement Program in January 2008. For more information, see my final summative report (artifact 6 on page 45) and the corresponding reflection (on Teaching-as-Research, page 11).

4) Development of a teaching and learning portfolio. Creating this document has helped me reflect on my experience as an educator.

Teaching Portfolio 1 P. X. Rault

http://www.delta.wisc.edu/

Teaching and Learning Timeline

The following timeline is to help the reader of this portfolio navigate through my courses and activities at UW-Madison, where I was a graduate student and TA from Fall 2003 to Summer 2008.


Mathematics TA training

Calculus (Math 171, 221, 222)

Wisconsin Emerging Scholars (Math 171, 217, 234)

Delta courses & activities

Elementary School teacher training (Math 130)

Delta internship (Math 320: Linear Algebra and Dif ferential Equations)

Delta-math TA training ("Participation in the Delta Learning Community")

2003 2004 2005 2006 2007 2008 2009

Teaching and Learning Philosophy

“My first experience teaching was inefficient and unsatisfying.”-page 4

“Convinced that I could study both mathematics and teaching, I broadened my educational techniques through the interdepartmental Delta / CIRTL

program in Research, Teaching, and Learning”-page 4

“I want my students to remember their math class for the rest of their lives, as a fun foray into abstract thinking. That is the unsurpassed joy of the ultimate puzzle, teaching.”

-page 5


Teaching and Learning Philosophy

"From the very beginning of his education, the child should experience the joy of discovery." -Alfred North Whitehead

I believe that active-learning is like solving a puzzle. Avoiding frustration while solving a puzzle involves three key steps: (1) thinking deeply, (2) having a guide, and, as Whitehead put it, (3) experiencing the joy of discovery. The ultimate puzzle is active-teaching.

My first experience teaching was inefficient and unsatisfying. As a graduate student at a major research institution I was told not to prepare for discussion sections and instead to focus wholly on research. My every impulse was telling me to master teaching as I had research. Answering questions on the spot at the board was unsatisfying to both students and myself, due to occasional long solutions which instilled illusions of difficulty. I longed to think deeply, have guidance, and find joy through discovery to this problem of teaching. I longed to teach as I had been taught as an undergraduate at a liberal arts college, through ample interaction with teachers and fellow students. Fortunately I have since resolved these problems.

When I taught for the Wisconsin Emerging Scholars program1 I realized my dreams. Students thought deeply on problems that facilitate learning-through-diversity2 of the students’ strengths. I thought deeply about how to challenge strong students without demeaning weak students, and was guided via books and peers to discover that instead of assigning a breadth of many problems I could provide depth through “optional” problem parts (cf. [1] for an example worksheet). Making “ASK QUESTIONS” the class motto, by encouraging questions of teachers and fellow students, formalized my deep rooted belief that student-teacher interaction is paramount in resolving misunderstandings. This belief is so strong that in Fall 2007 I implemented a study of interaction-maximizing-techniques for the opposite, large lecture, setting, the results of which I plan to present at the 2008 Joint Meetings of the AMS and MAA (cf. [1]).

Convinced that I could study both mathematics and teaching, I broadened my educational techniques through the interdepartmental Delta / CIRTL program in Research, Teaching, and Learning (cf. www.delta.wisc.edu). Backward design3 and Bloom's Taxonomy4 were especially helpful guides in my first independent teaching experience: elementary school teacher preparation. Midway through this course I surveyed students and re-assessed course goals such as (1) good mathematical expression via writing and (2) the ability to think deeply through challenging word problems. I wrote a detailed reflection (cf. [1] for a copy) of the thought process with which I backward-designed the remainder of the course. The final exam was both a joyful crowning achievement and my most difficult task to date. I once created an overly long survey and learned the hard way to ask only the key questions I want to analyze. The time I put into taxonomizing the crucial course concepts before writing this exam payed off 1The WES program “provides motivated students with an opportunity to study calculus in a challenging, friendly, multicultural environment.” It is based on the principles of the Emerging Scholars program at UC Berkeley and UT Austin. For more information, see www.math.wisc.edu/~wes2“Faculty and students bring an array of experiences, backgrounds, and skills to the teaching and learning process. Effective teaching capitalizes on these rich resources to the benefit of all, which we call Learning-Through-Diversity.” (cf. www.delta.wisc.edu).3Backward design is a systematic process involving three main steps: “Identify Desired Results, Determine Acceptable Evidence, and Plan Learning Experiences and Instruction” (cf. [3]).4“Bloom's Taxonomy provides a hierarchy of levels of cognitive understanding .... [and] an assessment framework that allows individual test questions to address a specific level in the taxonomy” (cf. [2])



incredibly: student performance varied primarily by learning styles as opposed to perceived problem difficulty! This well-planned exam (cf. [1] for the exam contents) took less time to grade and analyze than my ill-planned survey took to analyze!

My strongest reason for teaching is to guide student research. Projects can popularize mathematics and give motivation to study it. I once assigned semester projects in calculus 3 to learn the application of mathematics to a subject of interest, giving the example that an internet search for “Skateboard Science” yields a website which explains the physics behind every trick in the book of skateboarding. During our poster day, students revealed their joyful enthusiasm about subjects like the Rubik's Cube, the Elementary Mathematics curriculum, and Game Theory to a half-dozen professors and TAs. This was my first step towards my goal of mentoring students in independent studies and publishable research.

Nobody is perfect, and even after years of teaching, my model teachers still experiment with new approaches. I too experiment with new approaches in my teaching in pursuit of the joy of success. I want my students to remember their math class for the rest of their lives, as a fun foray into abstract thinking. That is the unsurpassed joy of the ultimate puzzle, teaching. As Albert Einstein once said, "It is the supreme art of the teacher to awaken joy in creative expression and knowledge."

References:[1] Rault, P. (2007). Teaching and Learning Portfolio. www.math.wisc.edu/~rault[2] Kastberg, S. E. (2003). Using Bloom's taxonomy as a framework for classroom

assessment. Mathematics Teacher, 96(6), 402-405.[3] Wiggins, G. and McTighe, J. (2005). Understanding by Design. Association for

Supervision and Curriculum Development.


http://www.math.wisc.edu/~rault

Mathematics Education Research Statement

“Some learn the art of good teaching by example, characterizing education as a social science. Others validate their methods empirically via sound research. I do both”

-page 7

“In Fall 2007, I designed and implemented an IRB5-approved controlled study of various large lecture techniques which facilitate student-teacher interaction.”

-page 8

“I am interested in initially studying high school and college courses and eventually broadening the scope of my research to K-8”

-page 9

5IRB is the UW institutional review board for human subjects research.


Mathematics Education Research Statement

1. BackgroundMathematics education is both a science and an art. Some learn the art of good

teaching by example, characterizing education as a social science. Others validate their methods empirically via sound research. I do both, treating mathematics education as an empirical social science.

As an art and a social science, there is a saying “the best way to find out what makes a good teacher is to ask the good teachers” (cf. [1]). Many educators act as gardeners, giving student-plants free range while ensuring they get the nutrients they need to prosper. Others act as tomes of knowledge and culture which they pass to the next generation (cf. [1]). These opposite methods can produce similar results, which has led to the idea that “there are no proofs in mathematics education” (cf. [4]). The teacher's “special something” makes the difference.

In any science, empirical evidence is critical. While proofs are impossible without axioms, overwhelming evidence was sufficient to convince the world of Newton's Laws. Later Einstein disproved them, and presented enough evidence to convince scientists of his theory of relativity. Psychology was once linked to Freud's controversial artistic opinions but now has well-founded theories governing the limits of memory. Even mathematics was once an art, as is illustrated by the following passage from the popular math book Fermat's Enigma:

“Pythagoras observed that Egyptians and Babylonians conducted each calculation in the form of a recipe that could be followed blindly. The recipes, which would have been passed down through the generations, always gave the correct answer and so nobody bothered to question them or explore the logic underlying the equations. What was important for these civilizations was that a calculation worked – why it worked was irrelevant” (cf. [5]).

The study of Mathematics Education as a science is its early stages. Skeptics cite lack of scope and replicability. Proponents cite studies revealing inadequacies in current methods. A middle-ground may be found. For credibility within mathematics circles, the researcher must first learn the mathematics before commenting on how to learn it. For credibility within education circles, a study must be done by someone competent in educational research. For change in either circle, one must follow a criteria such as in [4] to ensure that a study is credible and applicable. I am motivated to breach this divide between educators and mathematicians.

As a teaching assistant, I participated in numerous projects in mathematics education research. To understand learning communities I collaborated with a master's student in mathematics education and performed informal surveys and studies. The first project which I designed and implemented employed techniques which facilitate interaction between teacher and students in a large lecture setting. I will address results of each project below.

2. Learning CommunitiesLearning Communities are usually formed informally as peer study groups. Studies

have shown that minority students in the college environment (e.g. ethnically and culturally), who otherwise have the same motivation and aptitude, have high attrition rates. The Emerging


Scholars Program, pioneered by Uri Treisman at UC Berkeley (now at UT Austin), is designed to provide a learning community for these individuals. The principles guiding everything from group sizes to student and teacher responsibilities have been and continue to be researched [2].

The applicability of an educational study varies based on differences between the study group and implementing group. In the 2005-2006 academic year I taught a year long course of precalculus / calculus for emerging scholars. In Spring a student in mathematics education began work on a masters thesis in my course. Our collaboration focused on group dynamics and learning styles. Group composition, sizes, and roles were varied to maximize learning. A taxonomy of worksheet problem types was created to accommodate for student learning styles and confront student weaknesses. Many problems had “if you have time” parts to encourage fast students to delve deeper into challenging problems instead of completing more standard problems. Surveys, course evaluations, and grades revealed an improvement in student learning and fulfillment.

Motivation to think abstractly is difficult but important to foster. In the summer of 2006 I created a 30 minute motivation-inspiring presentation for the first day of an emerging scholars course. I focused on “mathematics as puzzle solving,” relating the desire to solve Sudoku problems or create strategies in games to the desire of mathematicians to solve math problems. I was invited to give this presentation at a UW Madison Engineering and Chemistry TA training. In the Fall 2006 semester I went one step further in motivating students by asking emerging scholars to choose a subject of interest to them and give a presentation about it. Students gave short oral presentations to the class about Mathematics Education, Rubik's Cube, Game Theory, Programming, etc. Later they created poster boards about their projects and discussed them during a poster day with professors and graduate students. Students were inspired by this learning process. Since then, I have facilitated ongoing student projects to deepen motivation, and created a student-oriented webpage comparing Games, Puzzles, Magic, and Mathematics.

3. Large Lecture TechniquesInteraction between students and teachers is critical to learning. Student

misunderstandings can often be settled by a simple question, but can be detrimental if allowed to fester. Learning communities, such as peer study homework groups, encourage students to resolve misunderstandings amongst each other. In large lecture settings, in-class group work (e.g. in the Emerging Scholars program) is difficult. In Fall 2007, I designed and implemented an IRB6-approved controlled study of various large lecture techniques which facilitate student-teacher interaction.

In Linear Algebra and Differential Equations I designated two discussion sections as control sections and two as research study sections. A pretest guaranteed that the research sections were not predisposed to do better. The controls were taught in the usual manner of questions and answers, while the research sections were taught using three common techniques for increasing student-teacher interaction. Student-response systems allow teachers to quickly gauge class understanding. Random contact (also known as names-in-a-hat) replaces the dynamic of 10% of students asking 90% of questions by a dynamic of all students asking questions. Pre-class discussion questions replaces the dynamic of vocal students choosing discussion topics by a democratic nomination and vote before class starts. Assignment grades and survey results were compared between sections. 6IRB is the UW institutional review board for human subjects research.


This project will be presented at the Joint Mathematics Meetings of the AMS and MAA in 2008. While this project was of small scope, may not be applicable to other settings, and is not a blind study, it does establish a case study which a teacher may choose to try in a course setting. The scientific method is similar to what I would use in the future to test a new teaching technique. I look forward to future projects of greater scale assessing and comparing teaching strategies.

4. Future ResearchIn future studies I will make a few changes to the methods in the latter project. This

project studied general students, while it would be more profitable to study different cultural groups. Teacher-specific bias may be countered by working in unison with other teachers and considering each course as one data point. A blind-study-administrator who selects research study groups will eliminate study bias.

Learning Communities and techniques to maximize Student-Teacher Interaction have many avenues for future study. The wealth of other ideas warranting further study are endless. I am interested in initially studying high school and college courses and eventually broadening the scope of my research to K-8. Specific research topics will be governed both by personal motivation to improve the system and by current trends in education.

Research targeting specific cultural groups will allow teachers to identify their strengths, to build on, and weaknesses, to work with. At UW Madison, the main groups unfortunately segregate: ethnic minorities, east coasters (called “Coasties”), and Wisconsinites (called "Sconies"). Research studies should isolate each culture's differences to maximize applicability to different situations.

Psychological limitations and strengths are well studied and may be applied to education. Schoenfeld discusses (cf. [4]) the limitations on working memory which prevent even the best mathematicians from quickly multiplying 379 and 658 with their eyes closed. As an undergraduate I took a year-long course in Psychology and would have pursued a minor if a second was allowed. I am very interested working jointly with a psychology researcher to understand learning.

Good teachers deserve to be asked how they do it. Specific methods such as visual models and graphing programs can be easily isolated to assess student understanding. Accounts of a teacher's artistic talent may be published as a case-study alongside other case-studies with similar results, as in [3]. This allows for documentation of mathematics education as an art.

In the near future, I hope to improve my knowledge of mathematics education techniques. I have taken courses rooted in science education on Teaching-As-Research7 and plan to pursue further mathematics-education-specific training. In any project I will use criteria for evaluating theories, like Schoenfeld's in [4], to maximize credibility. I look forward to collaborating with other educators to study cutting edge mathematics education techniques.

7 Teaching-as-research involves the deliberate, systematic, and reflective use of research methods to develop and implement teaching practices that advance the learning experiences and learning outcomes of students/participants and teachers/facilitators. (cf. www.delta.wisc.edu)



5. References[1]Andrews, G. E. (2001). “Review of Mathematics Education Research: A Guide for the

Research Mathematician by Curtis McKnight, Andy Magid, Teri J. Murphy, Michelynn McKnight.” The American Mathematical Monthly, Vol. 108, No. 3 (Mar., 2001), pp. 281-285[2]Asera, R. (2001). Calculus and Community: A History of the Emerging Scholars Program.

College Board. http://www.collegeboard.com[3]Friedberg, S. et al. (2001). Teaching Mathematics in Colleges and Universities: Case Studies

in Today's Classroom. CBMS, Issues in Mathematics Education, Volume 10.[4]Schoenfeld, A. H. (2000). Purposes and Methods of Research in Mathematics Education.

Notices of the AMS. Vol. 41, No. 6 (Jun./July 2001), pp. 641-649.[5]Singh, S. (1998). Fermat's Enigma. Anchor Books, 1st edition, 1998, ISBN: 0-385-49362-2.


Reflection on Teaching-as-Research

“This learning community dynamic provided constant feedback to students, first from other students, then from SAs, and then from the TA.”

-page 13

“The survey I implemented half-way through the course gave proof to my conjectures and light to my dilemmas.”

-page 13

“In my first lead-teacher experience, the final exam needed to be the crowning achievement.”-page 13

“I learned a lot in this 'getting my hands wet' experience with math education research.”-page 13


Reflection on Teaching-as-Research

DescriptionIn this reflection I will discuss several important artifacts from my teaching. Their

importance demands complete inclusion, while their length prohibits embedding in the text. I have resolved this dilemma by placing most of them in appendix 1.

After teaching for three semesters for the diversity-empowering Wisconsin Emerging Scholars8 (WES) program, I had an affinity for having students work on group worksheets with constant feedback in class. Artifact 1, on page 27, is such a worksheet. Standard problem types included “warmup,” “computation,” “application to real world,” “word problem,” “application of knowledge,” “theory/abstract reasoning,” “monkey wrench/challenge,” and “impossible problem.” Worksheets were made long enough that no student would finish but short enough that no student would be disillusioned. Often parts of problems would state “if you have time” so that students who are moving especially fast have a chance to delve deeper into one problem instead of skimming the surface of several problems. Of the above problem types, every student had their favorites. As students worked together, each student was able to show their strengths in their favorite problem areas while learning from the strengths of their classmates' favorite problems. Because students learned from each others' diverse skills and all had something to contribute, this was a good example of learning through diversity.

Teaching as lecturer of a course for elementary school teachers was my biggest challenge. In the syllabus at the start of the term, artifact 2 on page 29, I stated that teaching methods would be re-evaluated every block (approximately once a month) and thus new course timelines would be given periodically; included with this artifact are the timeline updates. All documents (including exam solutions and homework assignments) were uploaded to the course “Learn@UW” website for easy student access. At the end of the first two blocks I asked students to complete a Student Assessment of Learning Goals (SALG) online survey9 in which I included many personal questions about the effectiveness and use of techniques I was employing. I wrote a reflection about the experience, which included course changes. This reflection, which includes information about survey results, is artifact 3, on page 41. My fourth artifact, on the right, is a collection of excerpts from student evaluations concerning this course change.

The main grades in this course were projects (discussed in earlier reflections) and exams. There were three midterms referred to as “quizzes,” and one final exam. The final exam is artifact 5, on

8The Wisconsin Emerging Scholars (WES) program “provides motivated students with an opportunity to study calculus in a challenging, friendly, multicultural environment.” For more information, see www.math.wisc.edu/~wes. See my second reflection for more discussion.9http://www.wcer.wisc.edu/salgains/instructor/


➢ I know this is the first time teaching this class, and I have noticed great improvement throughout the semester. It was very rough/hard at first, but it is getting better (fair amount of HW/Quiz difficulty).

➢ The quizzes started off awful but have improved.

➢ Quizzes are getting better.➢ 1st one – not enough time, but has

gotten better.➢ The first quiz was ridiculously hard

then he made them better.➢ Homework is better than it used to be.

Quizzes are much more fair now.➢ Quizzes are made difficult but Patrick is

listening to comments and adjusting to fit circumstances.

Artifact 4: Student evaluations related to course changes

http://www.math.wisc.edu/~wes

, in which I employed aspects of Bloom's Taxonomy.10

Artifact 6, on page 45, is the summary of my research project for my course in Fall 2007. In this study, three teaching techniques geared toward large-lecture environments were compared against a control. This is my first official mathematics education research study.

Artifact 7, on page 54, is a handout for a joint mathematics and Delta TA training workshop to be held in the Fall of 2008. My goal in designing this workshop was to implement proven Delta-style methods into the mathematics department.

Reflections and AnalysisGroup problem solving via worksheets is an excellent way to run a class, if there is a low

student-teacher ratio. In my courses I had roughly 10 students and 2 undergraduate student assistants (SAs). Worksheets were created every day to be tailored to students' needs. Psycho-social factors created a need for constant group tweaking. Students were encouraged to ask and answer questions between each other to maximize their own learning. This learning community dynamic provided constant feedback to students, first from other students, then from SAs, and then from the TA. This Emerging Scholars program is well formulated with a strong research base. It is a great program to implement in any class environment where the student-teacher ratio is low.

Although it tested my endurance and my patience, I very much enjoyed teaching as lead lecturer a course for prospective elementary school teachers. I learned a lot from the experience of implementing the techniques I learned in my Delta / CIRTL courses. My syllabus was long: I provided concise answers to easy questions such as “when are my office hours?”, and detailed answers to important classroom dynamics such as “what are office hours used for?” The timetable updates of the syllabus integrated smoothly into the course. The survey I implemented half-way through the course gave proof to my conjectures and light to my dilemmas. This resulted in positive changes in the course which the students were grateful for (see artifact 4). Given the chance to redo the course, I would leave some techniques unchanged in the first half despite removing them in the second half; as described in artifact 3, these activities were necessary to change students conceptions of the course and how to present course concepts. The main change I would make is to clarify these activities. Writing the reflection (artifact 3) was instrumental to making informed decisions about the latter half of the course, as well as looking back on the experience for guidance in the future.

The Final Exam for this course was especially important. In my first lead-teacher experience, the final exam needed to be the crowning achievement. I used Bloom's Taxonomy to include questions distributed between computations and synthesis. In an orthogonal direction I distributed questions between each midterm's material. It was a difficult undertaking, but the results were excellent; for example it demonstrated varying learning styles as every problem was chosen as the “omitted problem” by at least one student.

Planning the research project was a big undertaking. I learned a lot in this “getting my hands wet” experience with math education research. Making everything crystal clear, before the project begins, to the IRB11 and to myself was very beneficial. Short surveys designed to ask a few key questions are much easier to analyze. This project has set a good foundation for comparison-based studies which I would like to do in the future.

Creating a TA training workshop was a good way of giving back to the UW community. This experience caused me to reflect about the most important lessons I learned as a teacher at

10Kastberg, S. E. (2003). Using Bloom's taxonomy as a framework for classroom assessment. Mathematics Teacher, 96(6), 402-405.11IRB is the UW institutional review board for human subjects research.


UW-Madison, which I felt that new TAs would profit by. The principal difficulty in creating this workshop was in dealing with politics within the mathematics department: there is both apathy among TAs and strong wildly varying opinions among professors.

Conclusions and PlanningThe Emerging Scholars program is excellent when well funded. I plan to seek funding

for such programs. When specialty programs are not possible, one can still improve the system with existing research, as I did in my Large Lecture Techniques project. It takes more dedication to create something from scratch, but I look forward to pursuing new methods and programs.

Teaching mathematics for prospective Elementary Teachers was a difficult blessing. I enjoyed the learning experience and look forward to teaching similar courses in the future.

Creating the final exam was a difficult task. In the future I will create exams before the start of the semester to guide the material I will introduce. It is difficult to plan everything in advance, so the exam will be revised shortly before it is taken. I learned this technique of “backward design” afterwards.

The research experience was enlightening and somewhat fruitful. While my data was mostly inconclusive, I did receive data showing that one of the methods was especially helpful to the students' learning in the class. I look forward to more in depth studies in the future, where I create new techniques and judge their beneficiallity.

The project to create a TA-training workshop was successful for two reasons. First, it caused me to reflect upon my most influential experiences as a graduate student. Second, the workshop will actually be used since Chris Pfund (Delta) and Bob Wilson (mathematics) have both agreed to run this workshop in Fall 2008. In the future, I plan to participate in similar endeavors.


Reflection on the Beauty of Math

“People solve Sudoku puzzles as a pastime, why not math puzzles?”-page 16

“The long-term projects combined with continuous feedback greatly improved students' willingness to think as a pastime.”

-page 17

“Thinking is a fun pastime; I want students to be fearless about intellectuality and grasp the joy of accomplishment.”

-page 18


Reflection on the Beauty of Math

DescriptionI love mathematics. I have heard many stories of students never having had a good

mathematics teacher, and many stories of students having one great teacher who made them love problem solving. I strive to be that teacher!

In the diversity-empowering Wisconsin Emerging Scholars12 (WES) program, students work in groups on difficult worksheet problems under the guidance of SAs (undergraduate Student Assistants) and TAs (as discussed in the previous reflection, paragraph 2 on page 11). Naturally, it is pivotal that students come to class prepared to think. For my third semester as a WES TA I created a motivational slideshow during the first week of class. I have included the entire slideshow, entitled “artifact 8,” in appendix 2.

Over the years I have had a few free class periods (after an exam, first day of class, etc) to introduce my students to math puzzles. Artifact 9 is an excerpt from a puzzle set I gave my students. I chose them hoping that students would see mathematics as puzzle solving instead of memorization. People solve Sudoku puzzles as a pastime, why not math puzzles?

While teaching a problem solving course for elementary school teachers I asked them to submit preliminary group-solutions to a problems by e-mail each day. Artifact 10 is an example of these solutions, which I copied for the students each day. In class we discussed pros and cons of each solution method and presentation style.

Students spend a lot of time surfing the web. Sometimes they find puzzle sites online, but they are more likely to when the puzzle site is run by their own teacher! Since I began inspiring students through puzzles and games I yearned to create such a site to (1) elaborate why one should study mathematics, and (2) talk about and propose some puzzles. Recently I created it, and artifact 11 is a snapshot.

Analysis and ReflectionOn the first day of class I found that Calculus 3 students were far more motivated than

precalculus students. I chose not to give this presentation, but instead to assign projects (which I discuss later). I was, though, invited to give this presentation at the Engineering and Chemistry TA Training.

My aim in giving my students puzzles was to give them: (1) a break from course material, (2) a chance to use abstract thinking without being graded, and (3) an impression that both puzzles and math are fun to do as a pastime.

12The WES program “provides motivated students with an opportunity to study calculus in a challenging, friendly, multicultural environment.” For more information, see www.math.wisc.edu/~wes


PuzzlesNovember 16, 2005

Puzzle 1 Philosophers100 philosophers are standing in a line. Each one has a hat,

either blue or yellow. Each philosopher can see every other hat, but not his or her own.

An angry king is forcing the philosophers to prove their worth. Each philosopher, one at a time in order, must say what color hat he or she has, or die a gruesome death.

If the philosophers are allowed to discuss a strategy in advance, what is the best possible strategy? That is, how many are they likely to save with what probability?

Puzzle 5 CoinsYou have 12 coins, all but one of equal weight. You also have a

balance scale, that will tell you which of two collections of coins is heavier.

If you want to figure out which coin is the heavier one, how many weighings do you need to do?Artifact 9: Excerpt from puzzles worksheet

Students loved the worksheets puzzles. It gave them a chance to think deeply without thinking about being graded. I have repeated this several times with more diverse puzzle worksheets; they especially loved working on puzzles on the first day of class!

Concerning problem types, students enjoyed most those problems for which they had the least questions to ask about before starting, presumably because they wanted to get as far as they could on their own. Some solo-minded students liked working on longer problems and getting continuous feedback. My most recent puzzle sheet was learning-through-diversity at its finest: each person was able to find a problem he or she wanted to solve! I have included this worksheet in appendix 2, as artifact 9b. However, I felt that these problems did not induce a long-term change in students' feelings towards problem solving. Long-term projects, involving writing solutions and gaining feedback, would be more profitable.

The long-term projects combined with continuous feedback greatly improved students' willingness to think as a pastime. Students improved their explanations and, probably because I started the course with some, they were fearless of difficult problems. For example, in


Math 130 - 2/1/2007 - Lecture 3Poison1) In order to win Rat Poison…1. You want to go second in order to win.2. If player 1 picks 3 pieces you, as the second player, should then pick 4 leaving 3 pieces in the middle allowing you to win.3. If player 1 picks 4 pieces you should then pick 3 pieces again leaving 3 pieces in the middle and again second player wins.4. If player 1 picks only 1 piece, you should then pick 1 piece too and strategize so that you are the player to leave either 3 pieces or 1 piece in the middle so that you win again. ***The ultimate objective is to be the second player and to leave either 3 or 1 pieces in the middle allowing player one no choice but to pick up the “poisoned” piece.***

2) First person always has the advantage of winning if he follows these rules:Pick up one counter. Let your opponent go. Pick up the same amount of counters as he did.Keep doing this and eventually he'll have to pick up the last counter.

4) Poison:With rat poison you always want to be the second player.If your opponent chooses four counters then you always take three leaving your opponent to take one, then you take one then your opponent is left with the last "poisoned" counter.4-3-1-1-1 = winIf however, your opponents takes three counters you should always take four counters, this will make leave your opponent with only one option, taking one counter, next you draw one once again leaving your opponent with the last "poisoned" counter3-4-1-1-1 =winFinally, if your opponent draws one counter you always want to take one counter, this leaves your opponent with three possible moves, they can take one more counter in which case you would take four and they would have to take one counter, you would then take one leaving them with the last "poisoned" counter.1-1-1-4-1-1-1 =winIf however, your opponent takes three on their second turn you should counter with drawing four counters leaving one counter left for your opponent.1-1-3-4-1 =winAnother option is if your opponent takes four on their second turn, you should then take one counter your opponent may draw one, followed by your removal of one and leaving your opponent with the last counter.1-1-4-1-1-1-1 = winFinally, the last option is if your opponent takes four on their second turn you can take three leaving your opponent with the last counter.1-1-4-3-1 =winI still haven't figured out why this works.

6) Go second and be not left with 1 or 3 counters.If your opponent takes 1 counter you have to take either 1 or 4. 3 will not work because you will be the one stuck with the 3 counters.Artifact 10: Excerpt from class worksheet: Poison Rat prelim. solutions.

five calculus courses I had trouble convincing students to think abstractly to understand that |x| and -x are equal whenever x is a negative number; students in this class took the time to understand this important concept and explained it wonderfully on the final exam.

Artifact 11 is a recent addition and I await student feedback to reflect upon. I will spur the process by mentioning it to students.

Conclusions and PlanningThe motivation presentation was well-received by the TAs in the TA training

symposium. I plan to give a similar talk in future group work courses. Motivating my students is important to me.

When only one class period is available, short activities can make a difference but the bulk of the problems should be kept short. Some more difficult problems should always be included for those who already enjoy abstract thinking. When I gave the problem sheet in appendix 2 (artifact 2b) on the first day of a Linear Algebra and Differential Equations course, the students loved the variety of problems. It was a success in Learning-through-Diversity: students were able to show their classmates how smart they were by choosing their favorite problem types. I will attempt the most successful approach whenever possible: ongoing problems with ongoing feedback.

To encourage students to study mathematics I will continuously improve my website to make it a resource for not only my own students but also the campus as a whole. Thinking is a fun pastime; I want students to be fearless about intellectuality and to grasp the joy of accomplishment.


Reflection on Undergraduate Research

“Students learned about their topics, effectively dispensed knowledge and excitement to peers and professors, and improved explanation and presentation skills.”

-page 20

“As an undergraduate at the College of William and Mary, a small liberal arts college in Virginia, I had high interaction with professors and partook in many small courses, independent

studies, and research projects – I wish to offer the same to my students!”-page 21

“I am happy to study a field where I can woo undergraduates with the magic of Fermat's Last Theorem and Elliptic Curves. I look forward to collaboration!”

-page 21


Reflection on Undergraduate Projects and Research

DescriptionIn undergraduate research students discover what mathematics really is: you must

problem solve, study, and collaborate. All skills are tested and potential results are high: abstract reasoning skills, good recommendations, a good career, and the cultural tag of genius.

But before an Undergraduate will begin research, he or she must first behold the beauty of math. Artifact 12 inspired me to assign class projects by answering the question: “what if I google a popular activity and math?” For instance, an internet search for “Skateboard” and “Science” yields a website devoted to demystifying the physics of skateboarding tricks; my artifact is a screenshot. Students chose a topic of interest (Rubik's cube, SONAR, game theory, binary, etc), delved into its relationship with mathematics, explained it in a class presentation, and created poster boards to facilitate discussion with invited professors.

The cross between “beauty of math” and “undergraduate research” is undergraduate projects. A result of another class project is artifact 13: a student solution to a mini project problem which I mentioned in my last reflection. This Hippo problem was: “Martha is studying hippo weights in Africa and wants to weigh 4 hippos. Her scale won't work for weights under 300 pounds, so they have to weigh hippos in pairs. They weigh the hippos in order of increasing combinations of weights, but on the last pair the scale breaks! How heavy was the last pair of hippos?” A later problem was “how heavy are each of the hippos?”, to which there where two different possible solutions...

Artifact 14 is aimed at advanced undergraduates: a short article about myself as an undergraduate at a liberal arts college. I look forward to collaborating with undergraduates just as my undergraduate and graduate research advisers worked with me.

Analysis and ReflectionThe results of the presentation and poster projects was incredible. Students learned about

their topics, effectively dispensed knowledge and excitement to peers and professors, and improved explanation and presentation skills. Students learned both applications of and abstract areas of mathematics. This experience was far superior to passively watching a Nova or assigning a popular math book to read because students gained confidence in their own topics and shared their diversity of knowledge with their peers. I have recommended it to my colleagues and created a webpage describing the process and listing student topics.

As discussed in my previous reflection, mini-projects involving the ongoing solution of problems in one problem type, coupled with continuous feedback by the teacher, create a great environment for improving understanding, explanations of understanding, and mathematical writing skills.


Once a student has become “mathematically mature”, he or she is called an “advanced undergraduate.” Perhaps these students will find a job in industry, go on to graduate school, or pursue some other abstract science. Regardless, it is a great experience for both student and teacher to work one-on-one in an independent study to do research or just to learn a topic. As an undergraduate at the College of William and Mary, a small liberal arts college in Virginia, I had high interaction with professors and partook in many small courses, independent studies, and research projects – I wish to offer the same to my students! Below is a screenshot of the online William and Mary viewbook, which should be an inspiration to students and to myself as their teacher. My own research lies in Arithmetic Geometry, the cross between Number Theory and Algebraic Geometry. Though stating results in the most general form requires some advanced mathematics, much can be done by undergraduates employing elementary techniques. I am happy to study a field where I can woo undergraduates with the magic of Fermat's Last Theorem and Elliptic Curves. I look forward to collaboration!


Hippo problemGiven six weight combinations: 312 kg, 356 kg, 378

kg, 444 kg, 466 kg, Error (breaks the scale).ASSUMPTIONS and notations:

*We will name the hippos A, B, C, and D, with A being the lightest, such that A<B<C<D (We can say this simply because of notation, since we choose that the lightest hippo be called A, the next heaviest be called B, etc).

Therefore, the two lightest hippos, A and B, weighed together would be the smallest mass combination, 312 kg. Likewise, hippos C and D together, the two heaviest hippos, would be the two hippos that broke the scale. We can make the following claims:

A+B= 312 (lightest)A+C= 356 (next lightest, since we know A+D, B+C,

B+D, and C+D will be heavier)A+D=?B+C=?B+D= 466 (second to heaviest, since we know that the

next heaviest, C+D, broke the scale, and B is the next lightest after C)

C+D= broken scaleNext, we make a comparison between the weights of

A+B and A+C. In both instances, hippo A is involved. The only thing that is changing is the second hippo involved. Since the weight difference between the two combined weights is 44 kg, one can say that hippo C, the heavier one, is 44 kg more than hippo B.

From there, we can establish what mass broke the scale. Since we know that Hippo C weighs 44 kg more than hippo B, and since we know that B+D=466, we can say that:

B=C - 44(C-44)* + D =466C+D=510*(C-44) is a substitution for B in the known equation

B+D=466The scale broke at 510 kg with the weights of hippos

C and D. Here’s a teacher solution to the problem at hand:

Artifact 13: Student solution to Hippo problem

Conclusions and PlanningProjects come in many forms and are commonplace in my classes. The choice of project

depends on the desired result: long-term continuous feedback on specific abstract problems, presentations about popular math, or other ideas to come in the future.

Research is the epitome of projects; every math major in their final year should have the chance to do a senior thesis. I will work hard to supervise and collaborate with undergraduates.


Artifact 14: Inspiring William and Mary Viewbook

Reflection on Approachability

“At the end of the year, the students gave me a touching Thank You card!”-page 24

“I set up the course for group discussions, made ASK QUESTIONS my motto, and conditioned myself to respond with glee to every question posed of me.”

-page 24

“Bonding makes students feel more free to approach you with misunderstandings on the material or questions about study methods, life, deeper material, etc.”

-page 25

“Every semester I will solicit phone numbers and provide copies for the whole class.”-page 25


Reflection on Approachability

DescriptionFrom Fall 2005 to Spring

2006 I taught a year long pre-calculus and calculus Wisconsin Emerging Scholars (WES) course (see my second reflection for a description). Near the end of the year, we had a breakfast party at the apartment of one of my student assistants (SAs), where we took a picture. At the end of the year, the students gave me a touching Thank You card! Together, these make up artifact 15, on the right.

The following semester I taught WES Calculus 3. To encourage friendship and collaboration between students and give them a way of contacting me in special situations, we all exchanged phone numbers. On one of the first Fridays of the semester we took advantage of the good weather to roast marshmallows at Picnic Point, a local picnic peninsula on the lake across from the State Capitol building, and took the outdoors picture (with the Capitol glowing in the background).

In Spring 2007 I was the lecturer for the mathematics course for prospective Elementary School Teachers. I set up the course for group discussions, made “Ask Questions” my motto, and conditioned myself to respond with glee to every question posed of me. Artifact 17 is a collection of comments from Teaching Evaluations related to my approachability to answering questions.


Artifact 16: WES Calculus 3, Fall 2006. A class outing at Picnic Point.

Analysis and ReflectionBoth WES Calculus

classes were small, which made out-of-class bonding feasible. The students, SAs, and I truly bonded and got to know each other. The experience is more productive at the start of a semester, but less feasible during a Wisconsin winter.

Bonding makes students feel more free to approach you with misunderstandings on the material or questions about study methods, life, deeper material, etc.

Exchanging phone numbers was especially helpful, though the course must be at a mature level for me to give students my phone number. Despite their diverse backgrounds, no students complained and most found it helpful. In Fall 2007 I provided students with the option of sharing their phone number and e-mail / instant messenger information with the rest of the class. I took special care not to pressure them, and 95% of students felt comfortable giving their name and e-mail.

The next semester my course was four times as large so I maximized the impact of in-class interaction by positively conditioning students to ask questions. Of the five classes which evaluations were taken from, a third of the comments on approachability (listed on the right) were from this course; it is working!

Conclusions and PlanningBonding with students at least once per semester is a good idea. I will have one or more

such bonding experience at a place like Picnic Point every Fall Semester, and will continue to seek out ideas for the Spring Semester. Every semester I will solicit phone numbers and provide copies for the whole class. Depending on my judgment of their maturity, I may also include my own phone number.

I will continue to condition myself to respond gleefully to questions, which conditions students to feel free to ask questions; the evaluations show that students appreciate it.


Teaching Evaluation Comments (related to approachability)

➢ I like how he makes it easy to ask questions.➢ Good at responding to questions.➢ Organized, he's easy to talk to about material.➢ He's a great TA, who is always ready to be helpful.➢ Very approachable.➢ Relaxed, thorough, always goes through salient points.➢ Lets us ask questions.➢ Patrick is friendly and very helpful. He explains things very well.

Keep up the good work.➢ He's very willing to answer any questions you have.➢ Friendly, enthusiastic.➢ Patrick is great with full explanations, and provides a great learning

environment (i.e. Takes questions and gives good answers). Thanks!➢ Interactive.➢ He is very approachable and he really encourages questions to make

sure we are understanding the material.➢ I feel at ease when in class. Tests are doable unlike some other TAs.

Makes an effort to answer students' questions.➢ He knows what he's talking about and makes us feel comfortable

correct his mistakes.➢ Optimistic about the material.➢ Very easygoing, fun to be around. Is open to suggestions.➢ Patrick is always willing to help us.➢ He is very approachable and available.➢ He is very approachable and helpful – makes an effort to make sure

everyone's up to speed.➢ He is very clear and understanding.➢ He is very relaxed and calming.➢ He always is willing to help, and gives different ways of explaining

things that are helpful.➢ I like most that he's given effort to improve and listen to our ideas.

Helpful. Patient.➢ Patient and kind. Helpful.➢ Patrick is really laid back and willing to listen to criticism.➢ Mellow.

Artifact 17: Excerpts from teaching evaluations, related to approachability.

Appendix 1: Teaching-as-Research Artifacts

The following artifacts are related to the reflection, on page 11, on Teaching-as-Research. Their length prevented their inclusion in the text.

Artifact 1: WES Worksheet. Page 27

Artifact 2: Syllabus with timetable updates. Page 29.

Artifact 3: Reflection on the first two blocks of class, and ensuing changes. Page 41.

Artifact 5: Final Exam (condensed). Page 43.

Artifact 6: A Research Project (final summative report for the Delta Program). Page 45.

Artifact 7: A joint mathematics-Delta TA training workshop. Page 54.


Artifact 1: WES Worksheet

Artifact 1: WES Worksheet - page 1


Artifact 1: WES Worksheet - page 2


Artifact 2: Syllabus with timetable updates

Artifact 2: Syllabus with timetable updates - page 1


Artifact 3: Reflection on the first two blocks of class, and ensuing changes

First two block of class:

Items I wanted the students to get out of the class

1. Deep understanding of the concepts of elementary-school-level mathematics.2. Self motivation and confidence to learn the subject on their own in the future.

Methods used: A denotes use primarily in 1st block, B primarily in 2nd block, and C primarily in 1st half of 3rd block. Note: Block1 was chapters 1-2, block 2 was chapters 3-4, and block 3 was chapters 5-6.

1. (A) Self-motivated use of teacher-suggested group roles (scribe, checker, facilitator (makes sure everyone is included), and moderator (makes sure nobody is dominating the discussion)

2. (A) Self-motivated use of teacher-suggested Learn@UW Discussion forums (I created a separate discussion groups for each class group).

3. (A) Self-motivated use of teacher-suggested Think-Pair-Square system of group interactions

4. (A,B) Rotating system for students to submit typed discussion questions about the reading.

5. (A) Discussion of questions in class

6. (A,B,C) Weekly Homework Assignments7. (A,B,C) Daily textbook reading assignments8. (A,B,C) In-class thinking problems9. (A,B,C) At-home write-ups of in-class problems10. (A,B,C) Teacher at-the-board lecturing to tie together

concepts11. (A,B,C) Group style discussion, in groups of 4

(sometimes 3 or 5).12. (A,B) Difficult problems, many of which were

intangible and not closely related to the subject at hand (e.g. algorithms, proofs).

Some changes over time, and my views of what happened.1, 2, and 3 where group facilitating methods which I tried instilling but did not want students to get bored with. Many students quoted them in discussion with their group mates as to how things should work. Number 2 was a big flop, since very few groups had more than one person willing to post something.

4 and 5 were slowly phased out, as it became clear that (1) the discussion questions did not teach them what the class was supposed to teach them: depth of elementary mathematics, and (2) the time was not available. In stage 3 it was removed completely because I had substitutes for a week.

6 through 11 I felt good about, and kept.

12 was lessened significantly at the suggestion of Professor Lempp, who sat-in on my course during the second block.

Student evaluations of these methods

On the week I was away at a conference, while my students where comparing me with my substitutes, I had them complete a UW Student Assessment of Learning Goals (SALG) survey. 39 out of 46 completed it without need for a reminder. Within it, questions similar to the following were asked with the following results. The format is Average, Standard Deviation.

A. How much did each of the following aspects of the class help your learning? (scale of 1 to 5: No help A little help Moderate help Much help Very much help)B. How often have you implemented / used the following aspects of the class to help your learning? (scale of 1 to 5: Very often, Often, Sometimes, A Little, Never)

Note that the scales are inversed (in the first, a 5 denotes “Very much help” whereas in the second a 5 denotes Never). Note also that I had no hope that 1,2, or 3 would be favorable, but rather implemented them to help those who otherwise feel unable to function in their group. I believe that the numbers reflect the percentage of people who feel this way, hence I should have included such a question to see a correlation.


mailto:Learn@UW

1. Group Roles (scribe, checker, moderator, and facilitator)

A: 1.74 (1.19)B: Average = 2.97, S.D. =

1.19

2. Learn@UW Discussion Forums 1.61 (1.1)Average = 4.56, S.D. = 0.81

3. The Think-Pair-Square system 2.15 (1.14)Average = 3.39, S.D. = 1.2

4. Discussion of questions that students type at home

2.31 (0.94)

Average = 3.9, S.D. = 1.01

5. Thinking up questions during reading

2.33 (0.92)Average = 3.56, S.D. = 1.01

*6. Homework problems 3.54 (1.08)Average = 1.64, S.D. = 0.7

*7. Reading the textbook 3.49 (1.06)Average = 1.87, S.D. = 0.94

*8. In-class problems3.13 (1.09)

x9. At-home write-ups of in-class problems

2.33 (0.89)Average = 2.64, S.D. = 0.97

x10. The teacher's at-the-board lecturing

2.84 (1.15)

*11. Group-style discussion3.34 (1.08)

12. How often do you have group discussions outside of class:

Average = 3.92, S.D. = 1.05

Any items scoring above a 3 on the A-scale or below a 2 on the B scale I marked with a *. These are items the students feel help. Any items which I feel strongly about which did not receive a *, I put an x next to.

ConclusionsIn light of the fact that I have been spending overly much time in areas that may not be of much good, and have areas where I would like to experiment further on my teaching, I will drop 1-5 and 9 for the second half of block 3 and all of block 4.

I will also work to improve my at-the-board lecturing (10).

I will focus the class on the following: 6. Homework problems, *7. Reading the textbook, *8. In-class problems, x10. The teacher's at-the-board lecturing, *11. Group-style discussion.


Artifact 5: Final Exam (condensed)

Final Exam – Spring 2007 -- Math 130100 Points

Name:

Section (circle): 9:30 1:00

Do FIVE out of the SIX problems. Put an X through the question you do not want graded (or do not write anything). If every problem is solved and no X is placed, then problem 6 will be left out.

You have one hour and fifteen minutes. Show all your work and make sure your explanations are crystal clear. If you need a clarification, raise your hand.

Five minutes before the end of class I will ask everyone to stay in their seats until the end, so that everyone may concentrate.

No calculators are allowed.

2) Estimation [20 points]

Estimate the fraction below using the standard method from class. Give an integer upper bound and an integer lower bound whose difference is at most 5. Show how you found them, and explain why you know they are smaller or greater than the following fraction.

617 55

3) Divisibility and PrimesThe following is a divisibility test for 13:

i) Let a be the one's digit of your number, and let b be your number with one's digit chopped off. Your number is 10 x b + a.ii) Your number is divisible by 13 exactly when b + (4 x a) is divisible by 13.iii) Let b + (4 x a) be your new number, and repeat the above steps until your number is less than 50. Note: the only multiples of 13 less

than 50 are 13, 26, and 39.

a) Demonstrate this test for the number 104 [5 points]

b) How would you check if a number is divisible by 39, using divisibility tests? Explain why your method works, and demonstrate it for 104. [5 points]

c) Using any method, factorize 4844. [10 points]


4) Critical Thinking Explanationsa) Absolute Values [10 points]

i) For any positive integer X, two of the following three are equal. Which two? Explain.X -X | X |

ii) For any negative integer X, two of the following three are equal. Which two? Explain.X -X | X |

b) Student Solution [10 points]You've given the following word problem to your students:

Jack and Jill's tourist shop sells T-Shirts for $10 each. Jack reduced that price by 20% today because they weren't selling well. Now they're selling great, so Jill will increase prices tomorrow by 10%. How much will a T-Shirt tomorrow?

Your student Julie gets correct answer, but mixed up the percents. Here is her solution:

10 percent of $10 is $1. A 10% increase is $11.20 percent of $11 is $2.20. A 20% decrease is $11 - $2.20 = $8.80.

Explain why this solution is wrong. Also, explain why for any other combination of prices and percentages you can mix up the percentages and still get the correct answer.

5) Computations (remember to show your work!)a) Rates [5 points]One hose fills a swimming pool in 10 hours, while the other hose fills the swimming pool in 40 hours. How long would it take to fill up the swimming pool using both hoses together?

b) Conversions [5 points each]i) Write 0.12 as a fraction in lowest terms.ii) Write the following as an exact decimal: 2

11

c) Simplify each of the following; any fractions should be put in lowest terms. You do not need to show any work. [1 point each]i) x2 - ( - xy ) - ( - x) ( x + ( - y ) )ii) 15 ÷ 30 13 39iii) 5 (3 / 7) + 2 (3 / 7)iv) ( 2 + (1 / 9) ) - ( 1 + ( 2 / 9 ) )v) -52

6) Prove that the square root of 2 is irrational [20 points]Hint: every rational number can be written as a ratio of two integers, and every integer has a prime factorization.


Artifact 6: A Research Project (final summative report for the Delta Program)

Large Lecture TechniquesFinal Summative Report

Patrick X. Rault

1. Abstract

The general purpose of the study is to test the effectiveness of certain educational techniques designed for large lecture courses. Two discussion sections of Math 320: Linear Algebra and Differential Equations at UW-Madison, were taken as control sections and taught normally. Two other sections were taken as research sections and taught using the following techniques to engage students in learning: student response systems, random contact, and pre-class submittal of homework questions.

Quantitative data on course-related grades and survey answers were taken and analyzed. Differences in grades between the control group and the research group were negligible, however students in the research group gave a significantly higher self-assessment of their own understanding of course material. Of the three techniques, students felt that homework question submittal was the most helpful learning aid.

2. Introduction, including:

a. The problem addressed

Large lecture environments often result in low student-teacher interaction, limited chances to resolve student misunderstandings of course content, and few possibilities for students to ask or respond to questions. We are interested in discovering which teaching techniques are best suited to improve student learning in the large lecture setting of a course of Linear Algebra and Differential Equations at UW-Madison.

b. What was known about the question/problem in the literature

The growing number of large lecture courses has decreased student-teacher interaction. Large numbers of students in a room combined with lack of time devoted to questions creates a dynamic in which (1) few people have enough ego to feel that their questions are worth asking and (2) the professor's apparent distance inhibits people from contacting him or her inside or outside of class. This encourages students to play a passive role in the classroom. Disliking this result, many educators have been spurred to create techniques geared toward maximizing active learning through interaction.13

Standard UW mathematics courses involve a large lecture by a professor (3 hours per week) paired with discussion sections by a TA (2 hours per week). Interaction between student and teacher is common but limited; the expensive Wisconsin Emerging Scholars (WES) program, which provides a learning community for students from diverse backgrounds, increases the TA contact hours to 6 per week to improve interaction through group work on problem sets14. My research instead concerns an inexpensive approach to math 320, “Linear Algebra and

13Buckley, Bain, Luginbuhl, & Dyer, Adding an "Active Learning" Component to a Large

Lecture Course, Journal of Geography, 103 (2004), p231-237.


Differential Equations,” where students have one contact hour per week with their TA, who in turn teaches four discussion sections of 24 students each. These sections share one important factor with large lectures: the short contact time and many discussion sections creates an impersonal environment15. Personal contact with each and every student is unwieldy.

A few techniques appropriate for use in this setting are student response systems, random contact, and homework question submittal. While others have used student response systems and random contact in non-mathematical settings, there is limited information about the use of these techniques in math courses in the literature. Submittal of homework problems is more common within mathematics, and is frequently practiced at UW-Madison. We will discuss these techniques further in the coming paragraphs.

Student response systems involve a multiple-choice question posed by the instructor to gauge understanding of a concept presented; buzzers at every seat allow students to respond to the question simultaneously. In standard lectures only a few students respond to understanding-oriented questions so the professor is unsure whether or not the response is representative of the class. We used a low-tech version of the student response system: three hard-paper cards are given to each student, with varying colors (red, green, blue) and the letters “A”, “B”, and “C” marked on them. The instructor may ask at any time a multiple-choice question and get a response which is easily judged from a distance; for example a sea of blue cards indicates that most students feel that “C” is the correct answer.

Random contact is a technique which is under-represented in the literature but is often a topic of conversation. Every student's name is placed in a hat, and periodically through the class the instructor draws a name. The student is given this chance to ask a question about the topic just presented, or, if he or she understood it well, a more-in-depth question. These random chances are naturally balanced with the usual on-the-go questions by ego-driven students. A “rainy day” scenario also applies: students wishing to “bounce” the opportunity to another student may do so with no impact on class assessment. Hence this technique does not put students on the spot to answer a question, but rather gives them the opportunity to ask a question. The instructor supplemented this technique with his own philosophy that asking questions is pivotal to understanding math; the class environment fostered the idea that not asking questions is worse than asking “bad questions,” by stressing respect of the idea that “anything which happens in the class should stay in the class” so students should not feel their peer's eyes upon them when they are asking for understanding.

In the UW mathematics department TAs conduct most discussion sections in one of two ways: (1) ask students to send them problems to go over in class via e-mail, or (2) answer students' questions on-the-spot during discussion section. While option (2) requires less effort by the TA, it is an inefficient and unrewarding use of precious contact time. Option (1) improves effective use of discussion time while requiring students to be more professional about their study time: a student who waits until the last minute to start homework may not have his or her questions answered. This stresses active-learning, one of the most successful learning methods in the literature, as students will take part in their own learning by planning study time and preparing questions effectively. We used option (1) along with an online “anonymous feedback” form to minimize the stigma of asking questions.

14Gomez, Wisconsin Emerging Scholars website, http://www.math.wisc.edu/~wes/

15Goodman, Koster, & Redinius, Comparing Biology Majors from Large Lecture Classes with TA-Facilitated Laboratories to Those from Small Lecture Classes with Faculty-Facilitated Laboratories, Advances in Physiology Education, 29 (2005) p112-117.


c. Overview of the accomplishments and challenges in your project

Students found the pre-class questions the most helpful technique by far. Both names-in-a-hat and student response systems were moderately helpful. No student found any of the techniques to be of no help.

3. How you addressed this problem, including:

a. Your project design

The general purpose of the study was to test the effectiveness of certain educational techniques designed for large lecture courses. Two discussion sections of Math 320: Linear Algebra and Differential Equations, were taken as control sections and taught normally. Two other sections were taken as research sections and taught using the following techniques: student-response systems, random-contact, and homework question submittal.

b. Key evaluation questions and methodology

Quantitative data was taken of group averages for quizzes, homework, exams, and answers to various SALG surveys (student assessment of learning gains). Measurements of differences between groups were recorded, as well as differences between answers to survey questions within the research group.

SALG questions asked of both groups were:1. How much did the way the class was taught, overall, help your learning?2. How much effort did you put into this course?

SALG questions asked of only the research group were:1. How much did each of the following aspects of the class help your learning?

a. Paper-card student response systemb. Names-in-a-hat random-contactc. Submittal of questions about homework (possibly anonymously) the day before discussion.

c. Examples of specific teaching and learning approaches and activities that you used to be an effective teacher for students with different backgrounds than your own

Learning-through-diversity is often difficult for me to implement but I believe that I succeeded in doing so with each of my three methods. Through names-in-a-hat, student response systems, and homework question pre-submittal, students were involved in asking and answering questions before and throughout class and were able to influence the planning, direction, and pace of the class. In standard discussion sections the most vocal students are poor representatives of the class and thus have some impact on the direction of the class but little impact on the pace. These systems allowed me, as the teacher, to acquire an unbiased view of average student understanding. This project has taught me that such a goal is attainable.

d. Examples of specific approaches you employed to develop and use learning communities


The three methods discussed in the previous paragraph also fostered a learning community environment by changing the class dynamic from lecture to discussion. In a community everyone has a role in its success, so the unbiased use of learning-through-diversity discussed above also helped to develop a learning community in which students felt equally important. Submitting homework questions prior to class was helpful in another way: in any community, it is the members' responsibility to come prepared to contribute. Over the course of the semester students were gradually convinced that looking over homework before class was essential to the functionality of the community, for two reasons: to have attempted the problems before discussing solutions on the board, and to allow the teacher to answer their questions in a well-prepared way.

4. Discussion of Evidence, including:

a. What evidence do you have that participants learned something?

Averages of quantitative data from course grades and surveys were calculated. Survey data was computed on a scale of 0 to 4, where 0 is “No help,” 1 is “A little help,” 2 is “Moderate help,” 3 is “Much help,” and 4 is “Very much help.” Grades are on a 100% scale.

Questions asked within the context of the survey were:1) How much did each of the following aspects of the class help your learning?

a. Paper-card student-response-systems.b. Names in a hat.c. Submittal of questions about homework (possibly anonymously) the day before discussion.

1) How much did the way the class was taught, overall, help your learning?2) How much effort did you put into this course?

Questions 1a, 1b, and 1c were only asked of the research group, while questions 2 and 3 were asked to both groups.

Whenever data was available for both groups, a t-test was used to compute the probability that these scores would occur at random. Thus a low score was a sign of a statistically significant result.

This data was used to create the following charts: Survey chart Key1) Student response (SR)2) Random contact (RC)3) Homework submittal (HS)4) Overall aid to learning (O)

○ t-test probability of random occurrence: 0.012

5) Effort put into learning (E)○ t-test probability of random occurrence:

0.3436) Vertical dashes designate the research group7) Diagonal dashes designate the control group


SR

RC

HS

O

E

0 1 2 3 4

Surveys (scale of 0 to 4)

Grade chart Key

1) Exam 1 grades (E1)○ t-test probability of random occurrence:

0.0122) Final Grades (F)

○ t-test prob. of random occurrence: 0.103) Vertical dashes designate the research group4) Diagonal dashes designate the control group

Note that control and research sections were treated identically until the first exam, so it is remarkable that they scored significantly (t-test probability of 0.012) better on the first exam. Concerning grades, we can only conclude that the use of our techniques correlated with negligible difference in student performance (measured through course grades).

Students in the research section felt that the homework question submittal (HS in the chart below) was most helpful to their learning. Comparing self-assessments of effort and understanding of students in the research and control sections (E and O, respectively, in the chart above), we conclude that students in the research section worked less but acquired a higher self-assessment of understanding of material.

b. How important was your particular approach to improving participant learning?

The specific approach was unimportant to the make-up of the project. There are many different methods which could have been used to turn this traditionally passive-learning lecture course into a more active-learning course. One objective of the project was to create a system for comparing innovative teaching methods against the norm.

c. Participant perceptions of their learning and feedback about your approach

Though I did not take qualitative data in this study, I will try to interpret what I observed of students. They enjoyed having an instructor who tried to improve the class in a caring manner. They did not take much advantage of random contact. Those who really desired to improve took my advice about looking at homework before class and sending questions about it in advance. Most forgot to or did not do this, however students rated this as between Much help and Very much help. The student response systems were popular, and most students did participate in this. However, students only felt that it was Moderately helpful. In practice, most students showed green cards (indicating that the pace was good), but when several students showed red (“speed up”) I received a good indication that I was working in too much detail and should speed up.

5. Lessons learned, including:

Here we will evaluate each of the methods discussed above. Students in the research section completed an end-of-class survey in which they rated each method on the scale of 0 to 4, where 0 is “No help,” 1 is “A little help,” 2 is “Moderate help,” 3 is “Much help,” and 4 is “Very


E1

F

0 20 40 60 80 100

Grades (% out of 100)

much help.” No student gave a 0 score to any of these methods.

Names-in-a-hat: Students often did not have a question to ask, or just wanted me to continue: “how do you do the rest of the problem?” or “can you do problem number X?” After some time I drew fewer names and other students started having questions to allow called-upon-students to bounce. More students ask questions when this delicate balance is achieved. Students found this system only a little helpful, giving it a 1.74 rating.

Pre-class questions: At first very few students (1-2) submitted questions for review. Some convincing was needed for my Tuesday class, so that students would at least look at their homework before discussion. After seeing other students participate, more decided or remembered to do so. On one occasion, I reminded students by e-mail and many more of them (8) submitted questions. Of the three teaching aids, students felt that this was the most helpful by far, giving it a 3.45 rating which is midway between “Much help” and “Very much help.”

Student-response system: Before implementation, my plan was to ask questions during computations. For example, I could ask “Is the answer to this computation A: 53, B: 76, or C: X2?” or “Is the best way to do this problem A: using the determinant, B: using row reduction, or C: using both?” It did not take long to realize that I had too little time in class to write down questions in such detail, and that the most efficient use of time was to write on the side board “A: going too slow, please speed up, B: I understand, your pace is good, C: I have a question or please slow down a little”. I was able to frequently ask them without much interruption, and students were able to quickly and easily lift up their cards to respond to them. Students found this moderately helpful, giving it a rating of 2.27.

a. What worked well?

The student response system was the most fluid in-class technique. It allowed me to keep a good pace for the class discussion. Pre-class questions allowed for a prepared and coherent course.

6. Address the following: (what changes and revisions, if any, would you suggest for the next time around?)

a. Project design and learning goals for participants

I suggest that a co-investigator, who is not an instructor for the course, conduct the grading of pre-tests and assigning of section roles (control versus research). This would allow for a more blind study. If possible, an arrangement could be worked out where one instructor grades while the other instructs.

For more validity, more sections should be involved. The oddity with grades (low t-test probability for the first exam) occurred as a result of one section doing phenomenally better than the three others. With more sections, this section could have been excluded as an outliers.

b. Your approach

The next time around I may tweak or remove the names-in-a-hat technique. It is difficult to implement and students do not appear to enjoy it.

The low-tech version of student response systems used in this project is best used to


allow the students to moderate the pace of the course. If repeated, I would use this system from the start. Another version of the student response system involves buzzers and a computer. With this system, instructors may put a question on a slide and immediately have a statistical distribution of all students' responses appear on his or her computer. The computer can also track the accuracy of these responses over the course of the semester, which allows for grading of this activity. This would be a possible, though costly, modification of this study.

The homework question pre-submittal technique was shown by this study to be profitable, so I will continue to use it.

c. Key evaluation questions and/or evaluation instruments

In this study, only quantitative data was taken. It would be possible to also take qualitative data, however this introduces more bias into the evaluation system.

7. Conclusions

These innovative techniques worked well in the setting of this class. Homework question pre-submittal worked best. Other techniques may be used with the same project plan.

8. Literature that informed your project

1) Buckley, Bain, Luginbuhl, & Dyer, Adding an "Active Learning" Component to a Large Lecture Course, Journal of Geography, 103 (2004), p231-237.

2) Gomez, Wisconsin Emerging Scholars website, http://www.math.wisc.edu/~wes/3) Goodman, Koster, & Redinius, Comparing Biology Majors from Large Lecture Classes

with TA-Facilitated Laboratories to Those from Small Lecture Classes with Faculty-Facilitated Laboratories, Advances in Physiology Education, 29 (2005) p112-117.

9. An appendix containing supporting materials (e.g., syllabus, activities, etc.)

a. Consent formUNIVERSITY OF WISCONSIN-MADISON

Research Participant Information and Consent Form Title of the Study: Large-lecture techniquesPrincipal Investigator: Bob Wilson (phone: (608) 263-5944) (email: [email protected])Student Researcher: Patrick Rault (phone: (608) 262-3600)DESCRIPTION OF THE RESEARCHYou are invited to participate in a research study about popular large-lecture techniques.You have been asked to participate because you are a student in Patrick Rault's lecture of Math 320.The purpose of the research is to validate and compare various techniques designed for large-lecture courses.This study will include Undergraduate students in Math 320.Research will consist of analyzing responses to online surveys and comparing grade data.WHAT WILL MY PARTICIPATION INVOLVE?If you decide to participate in this research you will be asked to complete online surveys and participate in in-class and homework activities.You will be asked to complete 1 survey.Your participation will last approximately 5 minutes per week for 15 weeks.


http://www.math.wisc.edu/~wes/

ARE THERE ANY RISKS TO ME?We don't anticipate any risks to you from participation in this study.ARE THERE ANY BENEFITS TO ME?Your involvement in this study will likely improve your understanding of the material from Math 320.HOW WILL MY CONFIDENTIALITY BE PROTECTED?While there will probably be publications as a result of this study, your name will not be used. Only group characteristics will be published.WHOM SHOULD I CONTACT IF I HAVE QUESTIONS?You may ask any questions about the research at any time. If you have questions about the research after you leave today you should contact the Principal Investigator Bob Wilson at (608) 263-5944. You may also call the student researcher, Patrick Rault at (608) 262-3600.If you have questions about your rights as a research subject you should contact the Education Research IRB at (608) 262-9710, [email protected] participation is completely voluntary. If you decide not to participate or to withdraw from the study it will have no effect on your grade in this class.Your signature indicates that you have read this consent form, had an opportunity to ask any questions about your participation in this research and voluntarily consent to participate. You will receive a copy of this form for your records. Name of Participant (please print):______________________________ _______________________________________ ______________Signature Date


b. Pretest


Artifact 7: A joint mathematics-Delta TA training workshop

Math Department Teaching and Learning Workshop

Chris Pfund, Patrick Rault, Bob WilsonDelta Program in Research, Teaching and Learning

www.delta.wisc.edu

Activities:1) Setting expectations: course syllabi2) Case study: changing sections3) Assessment of student learning: making the grade4) Case study: frustrated TA5) List of additional resources6) Workshop evaluation: 1 minute paper

What is the Delta Program?

The Delta Program is a research, teaching and learning community for faculty, academic staff, post-docs, and graduate students that will help current and future faculty succeed in the changing landscape of science, engineering, and math higher education. Through three core ideas: teaching-as-research, learning-through-diversity, and learning community, the Delta Program in Research, Teaching and Learning (Delta) supports current and future science, technology, engineering and math (STEM) faculty in their ongoing improvement of student learning.

You are invited to learn more about Delta and get involved in the many exciting opportunities that we offer. Choose the level of involvement that is right for you and your time constraints, whether it is one of our programs, graduate courses, internship and certificate programs, monthly Roundtable Dinners, and workshops, or other opportunities.

Feel free to browse our website to find out more about Delta: www.delta.wisc.edu -- be sure to contact us if you have any questions or want to talk about some way to get involved: [email protected]. We look forward to hearing from you!


mailto:[email protected]



SETTING EXPECTATIONS: COURSE SYLLABI(5 minute reading, 10 minute discussion)

First, spend several minutes skimming over the syllabi in the next few pages of this handout.

Next, split into groups of 3-5 people to discuss some of the following questions.

Discussion questions:1. What purpose does a syllabus serve? Does the purpose vary from course to course?

2. What are the main items that a syllabus should contain? Can there be too little (bullet points), or too much (paragraphs), information?

3. What common issues might the following groups of people face in your class: freshmen, first-generation college students, returning students who are over 30, disabled students, international students.

4. If the professor of your class has a syllabus, should you as a TA also have a syllabus?


CASE STUDY: CHANGING SECTIONS16

(10 minute reading, 10 minute discussion)

First, read through the case study alone. Then break into groups of 3-5 people.

16 Adapted from “Changing sections” in Friedberg, S. et al. 2000. Teaching Mathematics in Colleges and Universities: Case Studies for Today's Classroom. CBMS / AMS.


Discussion questions:1) To initiate the discussion, one person from each group should defend the positions of

Felicia and Otto.a) Should Felicia let Gil in or not? Put yourself in Felicia's position.b) Should Otto have advised Gil to drop the class? Take the role of Otto and give

his explanation to Gil of why he should drop down to precalculus.

3) Otto and Felicia are both successful teachers. Whose students benefit more in the long run, Felecia's or Otto's? What are the positive and negative aspects of each approach?

4) What are the appropriate goals of the first day of classes? How does one accomplish them?

5) Gil obviously has a passion for computer engineering. He may not be able to get a computer science major with a low grade in calculus. There is a shortage of engineers in the United States – how can you help solve this problem?


ASSESSMENT OF STUDENT LEARNING: MAKING THE GRADE17

(5-10 minute work & group discussion, 10 minute class discussion)

As a TA, you will be asked to grade homework, quizzes, and exams. The professor may provide guidelines but will leave most decisions to you.

You will likely be a teaching assistant for Calculus I. Grade the sample student work below first on a 10-point scale, as if they were problems on a quiz or an exam, and then on a 3-point scale, as if these were problems on the homework. Write your scores next to the student work in the appropriate column. (5-10 minutes)

If you finish early, look at the discussion questions on the next page.

17 Adapted from “Making the grade” in Friedberg, S. et al. 2000. Teaching Mathematics in Colleges and Universities: Case Studies for Today's Classroom. CBMS / AMS.


Discussion questions: 1) Does your grading scheme give A students a 90, B students an 80, C students a 70, and so

on? Is this important? Why or why not? If not, what is a fair way to curve the class?

2) Discuss different grading techniques which help ensure accuracy, consistency, and fairness, such as:

a) writing an accurate scoring rubric for each problem on a quiz or exam and keeping detailed notes of deductions made for various errors;

b) grading problem 1 for each student in the class before going on to grade problem 2 (on a test or homework), as opposed to grading each student's paper in sequence;

c) trying a “positive” approach (counting up from zero) to grading of assigning points to a student who correctly completes each step in a given problem instead of a “negative” approach (counting down from the maximum) of subtracting points for various types of errors.


CASE STUDY: FRUSTRATED TA18

(5 minute reading, 10 minute discussion)

“Since I had been careful to cover all the material in plenty of time for students to review, I thought that they would find the exam pretty straightforward. However, from their faces and comments as they handed the exams in it sounded like either I, or they, had misjudged:

'Man, that was hard. How'd you expect us to do that?'

'That was nothing – I mean nothing – like the homework.'

'I studied so hard, but it didn't help in the slightest on those problems. Next time, I may as well wing it for all the help studying gave me.'

My students were half freshmen, fresh from high school calculus, and half underclassmen. In class they were quiet and seemed bored. I tried to ask questions but no one seemed inclined to answer, so I decided not to push it. I was surprised by the lack of questions, but convinced myself that they were finding the course too easy.

My colleagues think I do a good job teaching—so, I don’t know what’s wrong. Students these days don’t know how to take notes and study. They just don’t get it.”

In a group of 3-5 people, discuss and write your answers to the following questions.

Discussion questions:1. How could he have avoided being so shocked at his students' performance on

the exam? What signals were there? What led him to misinterpret them?

2. How could the instructor have ascertained the level of his class's background?

3. What could the instructor do to improve student learning?

4. How could he find out whether learning has been improved?

18 Adapted from (1) Handelsman, J., S. Miller, C. Pfund. 2007. Scientific Teaching. W.H. Freeman & Co., New York. and (2) “Making the grade” in Friedberg, S. et al. 2000. Teaching Mathematics in Colleges and Universities: Case Studies for Today's Classroom. CBMS / AMS.


CONNECTING TO TEACHING AND LEARNING RESOURCES

UW-Madison & Mathematics TA resourcesDelta: http://www.delta.wisc.edu Mathematics TA Resources http://www.math.wisc.edu/~daughert/TAsite/Wisconsin Emerging Scholars program www.math.wisc.edu/~wesUW policy and information on cheating http://www.journalism.wisc.edu/bateman/index.htmUW TA resources http://www.ls.wisc.edu/TAresources.htmWPST: http://scientificteaching.wisc.edu/ Teaching & Learning Excellence@UW-Madison: http://www.provost.wisc.edu/tle/

Natioanl Mathematics Teaching OrganizationsThe Young Mathematician's Network http://concerns.youngmath.net/Mathematics Assoc. of America, TA handbook www.maa.org/programs/tahandbook.html

Courses and conferences in mathematics and science educationMathematics department Seminar in Mathematics Education:

http://www.math.wisc.edu/~wilson/Courses/Math903/current.htmlMathematics Education department graduate courses: 810, 811, 812Delta courses: College Classroom, Diversity in the College Classroom, etc.UW Engineering Teaching Improvement Program: http://www.engr.wisc.edu/services/elc/tip/UW Teaching and Learning symposium http://www.learning.wisc.edu/tlsymposium/

Nonstandard mathematics courses130, 131, 132 (create exams, full assessment responsibility, independent teaching, teach future K-12 teachers)101, 112, 113 (independent teaching, common course exams)WES 171, 217, 221, 222, 234 (6 / week of intense teaching one section of calculus, group work)Satellite calculus (indepndent teaching)Coordinator positions (create exams, full assessment responsibility)

Educational research journals and databasesERIC - Educational Resources Information CenterJSTORJournal of Research in Science Teaching Educational Studies in Mathematics Educational Technology, Research and Development Journal of College Science Teaching New Directions for Teaching and Learning Journal of Women and Minorities in Science and Engineering 2005 NSF Report: Women, Minorities, and Persons with Disabilities in Science and

National Center for Case Study Teaching in Science: Science Education Journals: http://ublib.buffalo.edu/libraries/projects/cases/journal.htm

Online teaching and learning resourcesTomorrow's Professor mailing list http://ctl.stanford.edu/Tomprof/index.shtml


http://ctl.stanford.edu/Tomprof/index.shtml

http://ublib.buffalo.edu/libraries/projects/cases/journal.htm

http://www.learning.wisc.edu/tlsymposium/

http://www.engr.wisc.edu/services/elc/tip/

http://www.math.wisc.edu/~wilson/Courses/Math903/current.html

http://www.maa.org/programs/tahandbook.html

http://concerns.youngmath.net/

http://www.provost.wisc.edu/tle/

http://scientificteaching.wisc.edu/

http://www.ls.wisc.edu/TAresources.htm

http://www.wisc.edu/students/saja/misconduct/misconduct.html

http://www.math.wisc.edu/~wes

http://www.math.wisc.edu/~daughert/TAsite/


University of Washington DO-IT: Disabilities, Opportunities, Internetworking and Technology: http://www.washington.edu/doit/CIRTL Innovative Programs Database: http://edweb2.educ.msu.edu/JHernandez/main.htm

References for studentsMathlab: http://www.math.wisc.edu/~mathlab/Math tutorial program: http://www.math.wisc.edu/~tprogram/Library exams: http://math.library.wisc.edu/reserve/Private tutors: http://www.math.wisc.edu/%7Epaulson/tutor.htmlYour syllabusYour office hours (three per week, or two plus by appointment)The Professor's office hours


http://www.math.wisc.edu/~paulson/tutor.html

http://math.library.wisc.edu/reserve/

http://www.math.wisc.edu/~tprogram/

http://www.math.wisc.edu/~mathlab/

http://ed-web2.educ.msu.edu/JHernandez/main.htm

http://www.washington.edu/doit/

WORKSHOP EVALUATION: 1 MINUTE PAPER

Take no more than a few minutes to write individually about the following two questions:

1) What new questions do you have about teaching and learning? 2) How will you answer them?


Appendix 2: Artifacts on the beauty of math

The following artifacts are related to the reflection, on page 15, on the beauty of math. Their length prevented their inclusion in the text.

Artifact 8: Motivation presentation. Page 69.

Artifact 9b: Puzzle set. Page 77.


Artifact 8: Motivation presentation.


Artifact 9b: Puzzle set.

(note: image for problem 4 was hand-drawn before photocopying)


teaching and learning portfolio - university … these rich resources to the benefit of all, ......

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